Adjoining batch Markov arrival processes of a Markov chain

作者：Mo, Xiao-yun; Xiang, Xu-yan; Yang, Xiang-qun^*

来源：Acta Mathematicae Applicatae Sinica-English Series, 2018, 34(1): 1-10.

DOI：10.1007/s10255-018-0724-3

• Markov chain
• batch Markov arrival process (BMAP)
• adjoining BMAP
• fixed constant batch
• independent identically distributed (i.i.d) random batch

A batch Markov arrival process (BMAP) X* = (N, J) is a 2-dimensional Markov process with two components, one is the counting process N and the other one is the phase process J. It is proved that the phase process is a time-homogeneous Markov chain with a finite state-space, or for short, Markov chain. In this paper, a new and inverse problem is proposed firstly: given a Markov chain J, can we deploy a process N such that the 2-dimensional process X* = (N, J) is a BMAP? The process X* = (N, J) is said to be an adjoining BMAP for the Markov chain J. For a given Markov chain the adjoining processes exist and they are not unique. Two kinds of adjoining BMAPs have been constructed. One is the BMAPs with fixed constant batches, the other one is the BMAPs with independent and identically distributed (i.i.d) random batches. The method we used in this paper is not the usual matrix-analytic method of studying BMAP, it is a path-analytic method. We constructed directly sample paths of adjoining BMAPs. The expressions of characteristic (D (k) , k = 0, 1, 2 center dot center dot center dot) and transition probabilities of the adjoining BMAP are obtained by the density matrix Q of the given Markov chain J. Moreover, we obtained two frontal Theorems. We present these expressions in the first time.

• 出版日期2018-1
• 单位湖南财政经济学院; 湖南文理学院; 湖南师范大学